Improved Small-Signal L2 Gain Analysis for Nonlinear Systems

TL;DR

This paper introduces a convex optimization approach to obtain tighter and more efficiently computed small-signal L2-gain bounds for nonlinear systems by reformulating the problem with linear matrix inequalities and novel error bounds.

Contribution

It advances previous nonconvex methods by developing a convex framework for tighter L2-gain bounds using LMI reformulation and improved triangulation error bounds.

Findings

Tighter upper bounds on system gain demonstrated numerically.

Convex optimization significantly improves computational efficiency.

Expanded applicability to nonlinear systems with combined storage functions.

Abstract

TheL2-gain characterizes a dynamical system's input-output properties, but can be difficult to determine for nonlinear systems. Previous work designed a nonconvex optimization problem to simultaneously search for a continuous piecewise affine (CPA) storage function and an upper bound on the small-signal L2-gain of a dynamical system over a triangulated region about the origin. This work improves upon those results by establishing a tighter upper-bound on a system's gain using a convex optimization problem. By reformulating the relationship between the Hamilton-Jacobi inequality and L2-gain as a linear matrix inequality and then developing novel LMI error bounds for a triangulation, tighter gain bounds are derived and computed more efficiently. Additionally, a combined quadratic and CPA storage function is considered to expand the nonlinear systems this optimization problem is applicable to. Numerical results demonstrate the tighter upper bound on a dynamical system's gain.